On the Stability of Equilibrium Paths Associated With Autonomous Systems

Abstract

The postcritical behavior and stability distribution on the equilibrium paths emanating from a divergence point associated with an autonomous system are studied within a state-space formulation. The analysis concerning the stability of equilibrium paths is based on the eigenvalues of the Jacobian evaluated at arbitrary equilibrium points in the vicinity of a critical point. Explicit conditions of stability and instability concerning the initial and postcritical paths are obtained through a perturbation approach. It is shown that at an asymmetric point of bifurcation an exchange of stabilities between two paths occurs in complete analogy with conservative systems. Similarly, a symmetric point of bifurcation involves a postcritical path which is totally stable (unstable) if the initial path is unstable (stable).